arXiv subjects

Young-Tak Oh

Publications and source records attributed to Young-Tak Oh.

24 records · Page 2Linked to original sources

arXiv2017

Shifted tableau switchings and shifted Littlewood-Richardson coefficients

We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {\em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of Schur $P$- and $Q$-function identities. We also demonstrate the relationship between the shifted tableau switching process and the shifted $J$-operation due to Worley.

math.CO↗

arXiv2016

The Universal Deformation of the Witt Ring Scheme

We determine the universal deformation over reduced base rings of the Witt ring scheme enhanced by a Frobenius lift and Verschiebung. It agrees with a q-deformation earlier introduced by the second author, for which we also give a simpler description. In the appendix we discuss a Witt vector theory for ind-rings which may be of independent interest.

math.RA↗

arXiv2015

Generalizing Witt vector construction

The purpose of this this paper is to generalize the functors arising from the theory of Witt vectors duto to Cartier. Given a polynomial $g(q)\in \mathbb Z[q]$, we construct a functor ${\overline {W}}^{g(q)}$ from the category of $\mathbb Z[q]$-algebras to that of commutative rings. When $q$ is specialized into an integer $m$, it produces a functor from the category of commutative rings with unity to that of commutative rings. In a similar way, we also construct several functors related to ${\overline { W}}^{g(q)}$. Functorial and structural properties such as induction, restriction, classification and unitalness will be investigated intensively.

math.RA↗

arXiv2006

$q$-deformation of Witt-Burnside rings

In this paper, we construct a $q$-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where $q$ ranges over the set of integers. When $q=1$, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. {70} (1988), 87-132). To achieve our goal we first show that there exists a $q$-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case $q=1$, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199 (1998), 703-732). Finally, we classify $\mathbb W_G^q$ up to strict natural isomorphism in case where $G$ is an abelian profinite group.

math.RA↗

arXiv2005

Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor $Nr$ from the category of special $\ld$-rings into the category of special $\ld$-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.

math.RA↗

arXiv2004

Generalized Burnside-Grothendieck ring functor and aperiodic ring functor associated with profinite groups

For every profinite group $G$, we construct two covariant functors $Δ_G$ and ${\bf {\mathcal {AP}}}_G$ from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor $W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, {\it Adv. in Math.} {\bf{70}} (1988), 87-132]. We call $Δ_G$ the generalized Burnside-Grothendieck ring functor and ${\bf {\mathcal {AP}}}_G$ the aperiodic ring functor (associated with $G$). In case $G$ is abelian, we also construct another functor ${\bf Ap}_G$ from the category of commutative rings with identity to itself as a generalization of the functor ${\bf Ap}$ introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, {\it Adv. in Math.} {\bf 81} (1990), 1-29]. Finally it is shown that there exist $q$-analogues of these functors (i.e, $W_G, Δ_G, {\bf {\mathcal {AP}}}_G$, and ${\bf Ap}_G$) in case $G=\hat C$ the profinite completion of the multiplicative infinite cyclic group.

math.RA↗